Cotangent space
From Academic Kids

In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. Typically, the cotangent space is defined as the dual of the tangent space at p, although there are more direct definitions (see below). The elements of the cotangent space are called tangent covectors. All cotangent spaces have the same dimension, equal to the dimension of the manifold.
Note that since the tangent space and the cotangent space at a point are both real vector spaces of the same dimension, they are isomorphic to each other. However, it is important to point out that they are not naturally isomorphic. That is, given a tangent covector there is no canonical tangent vector associated with it. The situation changes with the introduction of a Riemannian metric or a symplectic form in which case the added structure gives rise to a natural isomorphism.
For this reason it is important to maintain the distinction between the tangent space and the cotangent space. Many definitions are more natural on one space than on the other.
All the cotangent spaces of a manifold can be "glued together" to form a new differentiable manifold of twice the dimension, the cotangent bundle of the manifold.
Contents 
Formal definitions
Definition as linear functionals
Let M be a smooth manifold and let p be a point in M. Let T_{p}M be the tangent space at p. Then the cotangent space at p is defined as the dual space of T_{p}M:
 T_{p}^{*}M = (T_{p}M)^{*}
Concretely, elements of the cotangent space are linear functionals on T_{p}M. That is, every element φ ∈ T_{p}^{*}M is a linear map
 φ : T_{p}M→R
The elements of T_{p}^{*}M are called tangent covectors.
Alternate definition
In some cases, one might like to have a direct definition of the cotangent space without reference to the tangent space. Such a definition can be formulated in terms of equivalence classes of smooth functions on M.
Let M be a smooth manifold and let p be a point in M. Let I_{p} be the ideal of all functions in C^{∞}(M) vanishing at p, and let I_{p}^{2} be the set of functions of the form fg for f,g ∈ I_{p}. Then I_{p} and I_{p}^{2} are real vector spaces and the cotangent space is defined as the quotient space T_{p}^{*}M = I_{p} / I_{p}^{2}.
The differential of a function
Let M be a smooth manifold and let f ∈ C^{∞}(M) be a smooth function. The differential of f at a point p is the map
 df_{p}(X_{p}) = X_{p}(f)
where X_{p} is a tangent vector at p, thought of as a derivation. That is <math>X(f)=\mathcal{L}_Xf<math> is the Lie derivative of f in the direction X, and one has <math>df(X)=X(f)<math>. Equivalently, we can think of tangent vectors as tangents to curves, and write
 df_{p}(γ′(0)) = (f o γ)′(0)
In either case, df_{p} is a linear map on T_{p}M and hence it is a tangent covector at p.
We can then define the differential map d : C^{∞}(M) → T_{p}^{*}M at a point p as the map which sends f to df_{p}. Properties of the differential map include:
 d is a linear map: d(af + bg) = a df + b dg for constants a and b,
 d(fg)_{p} = f(p)dg + g(p)df,
The differential map provides the link between the two alternate definitions of the cotangent bundle given above. Given a function f ∈ I_{p} (a smooth function vanishing at p) we can form the linear functional df_{p} as above. Since the map d restricts to 0 on I_{p}^{2} (the reader should verify this), d descends to a map from I_{p} / I_{p}^{2} to the dual of the tangent space, (T_{p}M)^{*}. One can show that this map is an isomorphism, establishing the equivalence of the two definitions.
The pullback of a smooth map
Just as every differentiable map f : M → N between manifolds induces a linear map (called the pushforward or derivative) between the tangent spaces
 <math>f_{*}^{}\colon T_p M \to T_{f(p)} N<math>
every such map induces a linear map (called the pullback) between the cotangent spaces, only this time in the reverse direction:
 <math>f^{*}\colon T_{f(p)}^{*} N \to T_{p}^{*} M<math>
The pullback is naturally defined as the dual (or transpose) of the pushforward. Unraveling the definition, this means the following:
 <math>(f^{*}\theta)(X_p) = \theta(f_{*}^{}X_p)<math>
where θ ∈ T_{f(p)}^{*}N and X_{p} ∈ T_{p}M. Note carefully where everything lives.
If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward. Let g be a smooth function on N vanishing at f(p). Then the pullback of the covector determined by g (denoted dg) is given by
 <math>f^{*}dg = d(g \circ f)<math>
That is, it is the equivalence class of functions on M vanishing at p determined by g o f.
Exterior powers
The k^{th} exterior power of the cotangent space, denoted Λ^{k}(T_{p}^{*}M), is another important object in differential geometry. Vectors in the k^{th} exterior power are called differential kforms. They can be thought of as alternating, multilinear maps on k tangent vectors. For this reason, tangent covectors are frequently called oneforms.
Reference
 John M.Lee, Introduction to Smooth Manifolds, (2003) Springer Graduate Texts in Mathematics 218.
 Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) SpringerVerlag, Berlin ISBN 354042672.
 Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) BenjaminCummings, London ISBN 080530102X.
 Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0716703440.